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A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041.

%I A171692 #31 Mar 19 2022 02:32:41
%S A171692 1,1,10,1,1,56,246,56,1,1,246,4047,11572,4047,246,1,1,1012,46828,
%T A171692 408364,901990,408364,46828,1012,1,1,4082,474189,9713496,56604978,
%U A171692 105907308,56604978,9713496,474189,4082,1,1,16368,4520946,193889840,2377852335,10465410528,17505765564,10465410528,2377852335,193889840,4520946,16368,1
%N A171692 Triangle read by rows: absolute values of odd-numbered rows of A159041.
%H A171692 G. C. Greubel, <a href="/A171692/b171692.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A171692 T(n, k) = coefficients of (g(x, y)), where g(x, y) = n! * ((1-y)^(n+1)/(2*y)) * f(x, y, 0), with f(x, y, m) = 2^(m+1)*exp(2^m*x)/((1 -y*exp(x))*(1 +(2^(m+1) -1)*exp(2^m*x))).
%F A171692 From _G. C. Greubel_, Mar 18 2022: (Start)
%F A171692 T(n, k) = abs( A159041(2*n, k) ).
%F A171692 T(n, n-k) = T(n, k). (End)
%e A171692 Irregular triangle begins as:
%e A171692   1;
%e A171692   1,   10,     1;
%e A171692   1,   56,   246,     56,      1;
%e A171692   1,  246,  4047,  11572,   4047,    246,     1;
%e A171692   1, 1012, 46828, 408364, 901990, 408364, 46828, 1012, 1;
%t A171692 (* First program *)
%t A171692 f[x_, y_, m_]:= 2^(m+1)*Exp[2^m*x]/((1 -y*Exp[x])*(1 +(2^(m+1) -1)*Exp[2^m*x]));
%t A171692 Table[CoefficientList[SeriesCoefficient[Series[((1-y)^(n+1)/(2*y))*n!*f[x, y, 0], {x,0,30}], n], y], {n, 2, 20, 2}]//Flatten (* modified by _G. C. Greubel_, Mar 18 2022 *)
%t A171692 (* Second program *)
%t A171692 A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j,0,k}];
%t A171692 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] + (-1)^k*A008292[n+2, k+1], T[n, n-k] ]]; (* T = A159041 *)
%t A171692 A171692[n_, k_]:= Abs[T[2*n, k]];
%t A171692 Table[A171692[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 18 2022 *)
%o A171692 (Sage)
%o A171692 def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
%o A171692 @CachedFunction
%o A171692 def A159041(n,k):
%o A171692     if (k==0 or k==n): return 1
%o A171692     elif (k <= (n//2)): return A159041(n,k-1) + (-1)^k*A008292(n+2,k+1)
%o A171692     else: return A159041(n,n-k)
%o A171692 def A171692(n,k): return abs( A159041(2*n, k) )
%o A171692 flatten([[A171692(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022
%Y A171692 Cf. A008292, A060187, A159041.
%K A171692 nonn,tabf
%O A171692 0,3
%A A171692 _Roger L. Bagula_, Dec 15 2009
%E A171692 Edited by _N. J. A. Sloane_, May 10 2013
%E A171692 More terms from _Jean-François Alcover_, Feb 14 2014
%E A171692 Edited by _G. C. Greubel_, Mar 18 2022